chisquares

The chi-squared distribution is a fundamental concept in statistics, closely related to the normal distribution. It arises when examining the sum of squared standard normal variables. If Z1, Z2, ..., Zk are independent standard normal random variables, then the sum of their squares, Z1^2 + Z2^2 + ... + Zk^2, follows a chi-squared distribution with k degrees of freedom. The parameter k, representing the number of independent variables summed, is called the degrees of freedom.

The shape of the chi-squared distribution depends on its degrees of freedom. For a small number of degrees of freedom, the distribution is skewed to the right, with a peak near zero and a long tail. As the degrees of freedom increase, the distribution becomes more symmetric and begins to resemble a normal distribution. The mean of a chi-squared distribution with k degrees of freedom is k, and its variance is 2k.

The chi-squared distribution has numerous applications in statistical inference. One of its most common uses is in hypothesis testing, particularly in tests involving categorical data. For example, the chi-squared test for independence is used to determine if there is a statistically significant association between two categorical variables. Similarly, the chi-squared goodness-of-fit test assesses how well observed frequencies from a sample fit expected frequencies under a particular hypothesis. These tests rely on comparing an observed chi-squared statistic calculated from the data to a critical value from the chi-squared distribution to make decisions about the hypotheses.